A fundamental consumer issue in inkjet printing is the cost of replacement ink cartridges. While the retail prices of inkjet printers continue to erode, the per page cost of the ink replacement cartridges makes inkjet the most expensive of all the available desktop printing technologies.
The inkjet printer manufacturers have chosen to provide disposable inkjet cartridges which combine the electronically actuated nozzles with a small ink reservoir. This approach is justified by the fact that the "nozzles" have a limited life and must be periodically replaced. However, practice has shown that the volume of ink provided by the small reservoir in the disposable cartridges, is meager in comparison to the potential life of the nozzles, and that the volume of ink provided is limited for reasons other than nozzle life.
There exists technical ability to provide larger capacities of ink, and there is also economic incentive to provide replacement ink at more reasonable cost by leveraging the life of the nozzles. Thus, an aftermarket industry has evolved with the goal of providing more economical ink refilling options for inkjet printers. Many innovative means have been derived for refilling the existing inkjet cartridges or retrofitting larger reservoirs to the existing inkjet cartridges. Other approaches provide a second ink reservoir of larger volumetric capacity, usually in a stationary orientation, that transmits ink to the smaller reservoir of the translating ink cartridge. The process of interconnecting to the stationary reservoir has been accomplished in an intermittent fashion by parking the translating ink cartridge at an injector or pumping station for recharging. Alternatively, the stationary reservoir may be continuously connected to the translating ink cartridge by means of a flexible umbilical conduit and thus continuously recharge the smaller reservoir of the translating ink cartridge by utilizing either atmospheric or subatmospheric pressure differential. Both reservoirs in these dual reservoir systems must be recharged periodically, but at different intervals.
The most economic dual reservoir solutions take the form of extending an umbilical ink flow tube from the translating ink cartridge to a remotely located stationary reservoir. In such cases, the stationary reservoir may take the form of a disposable tank which can be exchanged several times during the useful life of the translating ink cartridge. In this way, the original nozzles of the manufacturer's inkjet cartridge can remain resident in the printer over the life of several exchanges of the remote tank. Since the stationary reservoir does not bear the expense of the electronics and nozzles of the resident inkjet cartridge, the stationary reservoir can be manufactured more economically and made available at considerably less expense. In this way, more reasonably priced consumables can be made available to the users of inkjet printers.
Such dual reservoir systems presently exist in industrial inkjet printer applications, but are noticeably absent amongst the array of desktop inkjet printers that are used in conjunction with personal computers in the office and home environments. There are three (3) problems that have precluded the use of dual reservoir configurations in such inkjet printers:
1) spatial inefficiency, PA1 2) cumbersome umbilical detachment means, and, PA1 3) costly umbilical configurations. PA1 where I=moment of inertia (in4) and y=distance to neutral fiber (in). PA1 w=0.50=width of the beam (in), f=0.10=height of the beam (in), and t=0.01=section thickness (in), PA1 substituting yields: EQU y=(0.1+0.01)-[((0.1).sup.2 +(2)(0.01)(0.10)+(0.5)(0.01))/((2)(0.5)+(2)(0.10))] in PA1 .thrfore.y=0.095833 in PA1 and EQU I=1/3[(0.01)(0.095833).sup.3 -(0.5)(0.10+0.01-0.095833).sup.3 -[(0.5-0.01)(0.10-0.095833).sup.3 ]] in.sup.4 PA1 .thrfore.I=3.40.times.10.sup.-6 in.sup.4 PA1 where I=moment of inertia (in.sup.4) and y=distance to neutral fiber (in). PA1 w=0.50=width of the beam (in), f=0.10=height of the beam (in), and t=0.01=section thickness (in). Substituting yields EQU y=(0.1+0.01)-[((2)(0.10).sup.2 +(4)(0.01)(0.1)+(0.5)(0.01))/((2)(0.5+0.2))] in PA1 .thrfore.y=0.089286 in PA1 and EQU .thrfore.I=[(2)(0.01)(0.10).sup.3 +(6)(0.10).sup.2 (0.01).sup.2 +(6)(0.10)(0.01).sup.3 +(0.5)(0.01).sup.3 ]/3-[((0.10+0.01-0.089286).sup.2 ((0.5)(0.01)+(2)(0.10)(0.01))] in.sup.4 PA1 .thrfore.I=6.03.times.10.sup.-6 in.sup.4. PA1 where DCF=dimensionless column factor=L .sqroot.(P/EI).
The spatial efficiency problem stems from the difficulty in integrating the umbilical and stationary reservoir into the printer chassis configuration in a compact manner. Provisions for the path and suspension of the umbilical and relative spatial requirements for the stationary reservoir usually mandate larger printer dimensions. Conversely, the quest for a smaller printer "footprint" is a major competitive factor in the desktop printer market which consequently distracts manufacturers from the consideration of a dual reservoir configuration. The ability to quickly and easily exchange the stationary reservoir is also required if such dual reservoir systems are to appeal to consumers. This quick change feature can only be achieved if the umbilical is also easily detachable and manageable without entanglement or ink leakage. Finally, the umbilical assemblage must be implemented in a simple and economic way that can render it disposable along with the inkjet cartridge.
The task of connecting a tubular liquid ink conduit between a translating ink cartridge and a non-moving supply reservoir is easily understood as shown by the prior art. However, it is a much more difficult task to integrate the path and motion of the conduit into a compact printer configuration. The detailed tasks of supporting, suspending, guiding, and controlling the path of the conduit is a serious engineering challenge. These tasks are even more acute problems when the goal of the configuration is to provide a disposable ink supply reservoir and a disposable inkjet cartridge, both of which are quickly and easily exchangeable.
It is important to recognize the mechanics of the umbilical loop motion in order to appreciate its inherent geometry problems. In FIG. 1, a gravity-oriented umbilical is shown attached to and extending from the left of a translating ink cartridge. The ink cartridge and umbilical are shown in two positions which might represent the width of the recording zone for the ink cartridge or some other boundary. The three essential elements of the umbilical include an unsupported upper portion extending leftward from the ink cartridge, a loop which traverses in the direction of the ink cartridge, and a lower supported segment that can be attached to a stationary reservoir. A marker m represents a point on the umbilical and shows the track of that point as the umbilical unrolls during the inkjet cartridge translation. The lower region of the umbilical is maintained in a flat shape because it is supported by a structural frame member which is straight. However, the upper section of the umbilical is not supported, and thus cannot be maintained in a straight orientation because of its flexibility. Note that the addition of a stationary support for the upper umbilical section would block the motion of the translating print cartridge and is thus impractical.
This unsupported upper umbilical portion takes the shape of a mathematical curve called a catenary. A catenary is known as the curve in which a flexible cable or cord will hang when supported at its two ends. The height of the catenary (y) at any position (x) along its span can be computed from the fundamental relation ##EQU1## where a=T.sub.h /w, T.sub.h is the horizontal component of the umbilical tension, and w is the weight per unit length of the umbilical.
The height of the loop, as shown by h.sub.1 and h.sub.2 in the two positions, is dependent upon the weight of the unsupported span of umbilical to its right, and the bending moment of the umbilical across the loop. The two vectors F.sub.1 and F.sub.2 are in equilibrium at any position of the umbilical where F.sub.1 represents the vertical reaction to the bending moment and F.sub.2 represents the left reaction component due to the weight of the unsupported span. As the ink cartridge moves leftward, the loop portion of the umbilical translates at a rate of one half the rate of travel of the ink cartridge.
The total motion of the loop would equal one half that of the ink cartridge (x.sub.1) if the radius of the loop would remain constant. However, the unsupported length of the upper umbilical decreases as the ink cartridge moves leftward. This reduces the length of the unsupported span and thus reduces the vertical umbilical weight component (F.sub.2). The result is less opposition to the bending moment of the umbilical, such that height of the umbilical (h.sub.2) is larger when the ink cartridge is at its left boundary. The traverse distance of the loop is thus only approximately equal to one half of the ink cartridge excursion, because the radius of the loop can be expected to gradually change as the ink cartridge traverses. This diagram also shows that a significant amount of space is required for the umbilical travel in the spatial region existing to the left of the leftward boundary of the ink cartridge excursion.
U.S. Pat. Nos. 4,677,448 and 4,757,331 (Mirusawa et al.) teach that control of the flexible conduit is a serious problem in configuring an appropriate printer structure. This prior art teaches that "it is required to reserve a large space for the flexible tube to move without trouble" and "since the tube is flexible, the locus of movement is not fixed but somewhat variable". "This large space dedicated to the movement of the flexible ink conduit has prevented the construction of smaller printers." For these reasons, the invention of Mizusawa et al. elected not to use an umbilical ink connection, but instead to provide an injection station. With this methodology, the translating ink cartridge is not connected to the stationary ink reservoir during the normal translation of the ink cartridge while printing or recording. Instead, the translating ink cartridge is transported to an injection station at certain intervals based upon ink exhaustion criteria, whereupon it is parked adjacent to an injector and pump which actively injects liquid ink from a larger volume ink reservoir. Such a configuration suffers the cost burden associated with pumps and injectors, as a result of not realizing a novel means for providing a more compact umbilical arrangement.
U.S. Pat. No. 5,473,354 (Arquilevich et al.) demonstrates a fundamental umbilical spatial problem as shown in FIG. 2. The umbilical assemblage is oriented in the plane of gravity underneath the translating ink cartridge and supported by a stationary frame member. The invention claims to "prevent the unwanted twisting of the fluid delivery tubes outside their vertical planes during carriage motion". This patent explains the obvious translation of the umbilical loop as the printhead traverses laterally, but does not show the orientation of the umbilical loop to the path of the printed media. Also noticeably absent is the identification and description of the relative orientation of the inkjet cartridge nozzles.
Current practice shows that the nozzles of inkjet printheads are oriented on the lower surface of the inkjet cartridge body in both atmospheric and subatmospheric type configurations. The term "lower surface" is interpreted in respect to gravity. The reasons for this are obvious and also include the need for gravitational support of the printed media in the gap adjacent to the nozzles. This printhead orientation causes a significant problem if the umbilical loop must also operate in the plane of gravity (vertical plane). In such cases, the umbilical must have sufficient length and travel in its operating plane, to prevent the loop from cutting across the plane of the media, as shown by the illustration in FIG. 3. The operational zone of the loop must exist leftward of the media path and requires a width equal to about one half that of the printhead excursion zone plus some spatial clearance for the loop. The umbilical operational zone is shown in FIG. 3 by the dimension (k.sub.1 +x/2). This illustration thus explains that the printer chassis would need to include provision for the significant extra space that is required for the motion of the umbilical loop. A printer which used this umbilical configuration would require at least a 50% wider footprint than the same printer without an umbilical.
It can be seen that the gravity-oriented umbilical suspension is not amenable to the goal of providing a compact desktop printer. The explanation in Arquilevich et al. teaches that spatial efficiency was not a goal of this arrangement, stating that "the actual length and configuration of the apparatus is not important provided that the ink-delivery tubes are sufficient to connect one or more ink sources to the moveable printhead".
U.S. Pat. No. 5,561,453 (Shibata et al.) is another example of the prior art which teaches an umbilical loop translation but fails to explain the relative orientation of the media path.
U.S. Pat. No. 4,684,962 (Orosawa et al.) shows an umbilical shape in FIG. 4 that more appropriately illustrates the shape of those actually found in practice. This prior art also fails to disclose the relative orientation of the media path.
Other prior art seeks to solve problems which are the aftermath of inadequate umbilical control. Shibata et al. teaches a solution to the potential problem of the conduit being collapsed by a "kink" or entanglement, as shown in FIG. 5. A custom profile flexible ink carrying conduit contains multiple chambers to allow fluid ink flow when the primary chamber becomes restricted as a result of collapsing or kinking. This prior art does not, however, teach how to support, guide, suspend, or control the path of the flexible conduit so as to initially prevent the problems of collapsing or kinking.
Arquilevich et al. teaches a solution to bundling groups of flexible tubes in the situations where multiple ink supply conduits are utilized, as in the case of multicolor printer devices. As show in FIG. 6, a thin membrane material is bonded over the periphery of several circular tubes such as to interconnect the individual tubes into a common structure which will prevent entangling. Hirosawa et al. also teaches a multiple conduit approach to help manage the movement of the flexible ink supply interconnections. The two tubes are made to move in parallel to each other by virtue of their exit orientation relative to the traversing ink cartridge carrier structure. However, neither Arquilevich et al. nor Hirosawa et al. proposes a solution to the case where only a single ink conduit is utilized. Furthermore, none of these inventions seeks to explain a solution to the problem of guiding, suspending or controlling the path of the flexible conduit assemblage. Nor does this prior art seek to minimize the spatial requirements of the umbilical.
U.S. Pat. No. 3,583,732 Dennis et al.) discloses a helically wrapped wire spring to enshroud and support a fluid (air) duct, and thus prevent the collapsing of the internal cavity of such a duct. Longitudinal wires are attached along the peripheral axes of the duct to render the duct inflexible. This arrangement lacks flexibility and would thus not be suitable for the task of providing a flexible conduit from a stationary reservoir to a translating ink cartridge.
U.S. Pat. No. 5,449,021 (Chikama et al.) teaches the use of a helically wrapped spring in conjunction with control wires, to provide a controlled flexible conduit motion in one plane. Corrugated metal strips are spot welded to the periphery of the helix along the longitudinal axis of the spring on two opposing sides as shown in FIG. 7. The helices along two sides of the spring are thus rendered inflexible in the plane formed by the two spot welded strips. Two control wires are threaded through the internal cavity of the conduit and terminated at the flexible end of the structure at positions which are orthogonal to the spot welded bands. Rotation of a pulley in the control end of the conduit then pulls one of the wires to create limited flexure in one plane. This technique is a complex and costly solution to the problem of providing a conduit that is flexibly controllable in one axis only. Such complexity and cost precludes its utilization in a disposable and economically sensitive inkjet cartridge application.
Many methods for interconnecting flexible liquid carrying conduits have been shown in the prior art, including those that clamp needle-like probes to a septum. U.S. Pat. No. 5,137,524 (Lynn et al.) shows a sliding collar to clamp a probe, which contains a needle, to the exterior surface of a septum after the needle has penetrated the septum. This is shown in FIG. 8. While this invention presents a means to couple two sections of flexible conduit, it does nothing more than provide a system of two possible states consisting of either a coupled or uncoupled flexible conduit. In the first, or coupled state, there exists a connection between the remote liquid reservoir and the receiving end (patient's vasculature), but little control over the suspension and path of the flexible conduit. In the second, or uncoupled state, the control of the conduit is lost completely.
The lack of a docking station, and lack of means to control the umbilical assemblage, are deficiencies which prevent earlier invented stationary reservoir systems from presenting a quickly exchangeable reservoir or a quickly changeable translating ink cartridge.
U.S. Pat. Nos. 5,369,429 and 5,367,328 (Erickson et al.) show stationary reservoir ink delivery systems where ink is delivered through one or more flexible tubes to translating ink cartridges. The individual tubes are affixed to cavities in the individual links of a link chain which provide the support elements for an umbilical assembly. At the reservoir end of the flexible tubes, the tubes enter the stationary reservoirs through an orifice which is both strain relieved and sealed to a plastic reservoir liner, such that the tubes are a permanently and inextricably attached to the remote reservoirs. FIG. 9 shows this reservoir and ink cartridge without the link chain. The process of exchanging the remote tank then requires that the tubes be individually unthreaded and rethreaded through the individual clamping stations of the chain link umbilical assemblage. Before such threading can take place, the support chain itself must be detached from the printer structure. The tedious nature of this threading and unthreading process, combined with the lack of an umbilical docking station, prevents the possibility of providing a quickly exchangeable reservoir.
Other prior art proposes to operate the umbilical in a plane that is perpendicular to the plane of gravity. U.S. Pat. No. 5,469,201 (Erickson et al.) presents two solutions to constraining the movement of an umbilical to one plane of motion. A preferred solution is highly flexible but unstable, while an alternate solution is excessively inflexible. In the prior art preferred embodiment shown by FIG. 10, plastic link chain is utilized as a flexible umbilical carrier and looped into a plane which is orthogonal to its hinge pins, such that it is self supporting in that particular plane. However, while the link chain is constrained to that plane, its movement within that plane is not predictable. This problem is resolved by the addition of a flat metal band which is mounted adjacent to and along the periphery of the link chain to aid in controlling the shape of the loop. The result is an umbilical assemblage that is self supporting in one axis which is perpendicular to the axis of gravity, but flexible in the other two axes. This umbilical assemblage can only be self-supporting when arranged within the printer configuration such that the axis of the chain link hinge pins is oriented within the axis of gravity. This restriction then presents an impediment to providing a compact printer configuration, as the plane of the looped umbilical cannot exploit a more vertical orientation. A more vertical umbilical orientation is a requirement to position the reservoir as closely as possible to the path of the translating ink cartridge, and thus to provide a compact printer configuration as shown by the invention herein. Further, since this combination of link chain and flat metal band is complex and costly, it precludes the possibility of providing an economically disposable umbilical.
An alternate embodiment of Erickson et al. '201 is explained which uses a "unitary piece of rigid, flexible material" that is described and shown as being in the shape of a rectangular channel made of plastic or metal. The channel carries a series of attachment mechanisms, which are spaced intermittently at 3 to 4 inch intervals along the length of the channel, and whose purpose is to support the ink supply tubes. Each of the individual attachment mechanisms supports a "closure used to secure the supply line", which is further described as a "pivoting clamp mechanism". The structure and operational principles of the "pivoting clamp mechanism" and the means for attaching it to the channel, are not disclosed. The elongate channel is further described as maintaining a "generally U-shaped" structure and is diagrammatically shown as maintaining its cross sectional channel shape continuously without deformation, around a 180 degree bend with a smooth radius. Two views of this channel and its ink tubes are shown in FIGS. 11 and 12.
The stiffness attributes of a structure, or conversely its flexibility attributes, are composed of two factors; shape and material composition. Any given material possesses a characteristic property called its elastic modulus, that uniquely dictates the degree of flexibility that a given shape can achieve when utilizing that particular material. For example, a 2".times.4" wooden stud made of Douglas fir such as used in residential building construction (a common two-by-four) can be compared in stiffness to a 2".times.4" stud which is made of steel. The Douglas fir possesses an elastic modulus of 1.9 million psi and the steel possesses an elastic modulus of 30 million psi. Thus, the steel stud is stiffer than the wood stud due to its significantly higher elastic modulus. The actual difference in stiffness between the two studs can be computed by simply comparing the ratio of elastic moduli. When checking this ratio, we can conclude that the steel stud will be exactly 15.79 times stiffer than the wooden stud. Conversely, we can also conclude that the wooden stud is 15.79 times more flexible than the steel stud. Thus it can be seen that a homogeneous material possesses a unique property, its elastic modulus, that quantifies its inherent flexibility regardless of shape. The description of a material that is both rigid and flexible, as explained in Erickson et al (U.S. Pat. No. 5,469,201), is therefore clearly a self-contradiction.
More important to the issue of flexibility is the shape of the cross section of a structural member. In the field of structural mechanics, a characteristic number can be computed for any shape that inherently predicts how flexible that shape will be when subjected to forces which are imposed along different axes. That property is known as the moment of inertia of a sectional shape. Unlike materials, shapes in themselves can be inherently flexible or inherently inflexible by virtue of this property. Further, the moment of inertia is axis-dependent such that a given shape may have a moment of inertia that can be different in one axis than in another. An example of this can be easily understood from analyzing a thin flat metal strip.
For this example, assume that the goal was to make a thin strip that was to be very flexible. Common sense would dictate a shape with a high ratio of width to thickness. As an example, the dimensions of 1/2" for the width and 10 mils for the thickness might be chosen, which are similar to the dimensions of a small metal ruler. The ratio of width to thickness is 50:1 in this case. The moment of inertia for this rectangular shape along its bending axis can be easily computed from the equations of structural mechanics, which are found in most engineering handbooks. The results of such a computation yields a moment of inertia of 0.041.times.10.sup.-6 in.sup.4 as shown below.
Moment of Inertia for a Thin Rectangular Section (across thickness)
Referring to FIG. 3A, I=wt.sup.3 /12, where I=moment of inertia (in.sup.4). Since w=0.50 and t=0.01, I=(0.50)(0.01).sup.3 /12, and I=0.041.times.10.sup.-6 in.sup.4.
Alternatively, the moment of inertia of the strip could be computed for the direction across its width. The results of that computation yields a section modulus of 104.166.times.10.sup.-6 in.sup.4 as shown below. These results show that the strip is tremendously more stiff across its width than it is across the plane of its thickness and explains why the strip is rigid in that plane. The stiffness of the strip in each of the two planes can be compared by taking the ratio of the moments of inertia in the two planes. This shows that the strip is 2,540 times stiffer in the width direction than in the thickness direction, irrespective of the material that is utilized. This explains why large ratios of width to thickness are chosen in the case where both lateral stability and a high degree of flexibility are required, such as in the cases of power transmission belts which must wrap around pulleys.
Moment of Inertia for a Thin Rectangular Section (across width)
Referring again to FIG. 3A, I=tw.sup.3/ 12, where I=moment of inertia (in4). Therefore, EQU I=(0.01)(0.50).sup.3 /12 and EQU I=104.166.times.10.sup.-6 in.sup.4.
Assume that this strip was deemed "too flexible" for a given application, and that the shape should be modified in order to make the strip more structurally rigid. One way to achieve more rigidity is to add a "stiffening rib" projecting perpendicular to the thin section of the strip, such that the shape of the strip looks like a "T" section. The data below shows the quantified increases in stiffness for added sections of various ratios of the strip width. For example, a flange depth of 20% of the strip width produces an 82.9 times stiffness increase, so that the strip is no longer easily bendable.
Moment of Inertia for a "Tee" Section
Referring to FIG. 3B, y=(f+t)-[(f.sup.2 +2tf+wt)/(2w+2f)] in and EQU I=1/3[ty.sup.3 +w(f+t-y).sup.3 -[(w-t)(f-y).sup.3 ]] in.sup.4
For the comparative case where flange depth=20% of beam width,
In like manner, comparative moments of inertia are computed for the cases of 10% and 30% flange-to-width ratios and shown in the following table:
______________________________________ Moment of Stiffness Sectional Shape Inertia in.sup.4 ratio ______________________________________ Thin rectangular 0.041 .times. 10.sup.-6 1.0 "T" w/10% 0.555 .times. 10.sup.-6 13.5 flange ratio "T" w/20% 3.40 .times. 10.sup.-6 82.9 flange ratio "T" w/30% 10.20 .times. 10.sup.-6 248.8 flange ratio ______________________________________
Another method that can be used to stiffen the strip is to add two "stiffening ribs" at the longitudinal edges of the strip, such that it cross section takes the shape of a channel.
The increase in rigidity due to the channel shape is also intuitively understood, but the examples below have been generated to quantify the increases in stiffness, for added sections of various ratios of the strip width. For example, a flange depth of 20% of the strip width produces a 147 times stiffness increase, as compared to the thin rectangular strip. The relative proportions of these shapes is shown in FIG. 13. It can be seen that the channel shape is an extremely effective stiffening shape, and produces more rigidity than the "T" shape, when using the same size envelope. This is the reason that the channel shape is commonly used for steel beams in the construction of commercial buildings.
Moment of Inertia for a Channel Section:
Referring to FIG. 3C, y=(f+t)-[(2f.sup.2 +4tf+wt)/(2w+4f)] in and EQU I=[2tf.sup.3 +6f.sup.2 t.sup.2 +6ft.sup.3 +wt.sup.3 ]/3-[(f+t-y).sup.2 (wt+2ft)] in.sup.4
For the comparative case where flange depth=20% of beam width,
In like manner, comparative moments of inertia are computed for the cases of 10% and 30% flange-to-width ratios and shown in the following table:
______________________________________ Moment of Stiffness Sectional Shape Inertia in.sup.4 ratio ______________________________________ Thin rectangular 0.041 .times. 10.sup.-6 1.0 Channel w/10% 1.0 .times. 10.sup.-6 24.4 flange ratio Channel w/20% 6.03 .times. 10.sup.-6 147.0 flange ratio Channel w/30% 17.70 .times. 10.sup.-6 431.7 flange ratio ______________________________________
It can thus be seen that the rectangular channel shown by the alternate embodiment of Erickson et al. '201 is a structure that is inherently rigid, and thus difficult to bend whether it be made from plastic or metal. Since the forces required to bend such a structure must be continuously provided by the transport motor which translates the printhead, such motor will require considerable extra power for the task of continuously bending and unbending a channel structure during the translational printing process. This can be shown by the analysis below which compares the force required to bend a thin flat strip with those to bend a channel of the same thickness over onto itself into the shape of a loop.
The channel illustrated in the Erickson et al. '201 patent diagrammatically shows lateral flanges that are proportioned to the channel width by about a 20% ratio. It is understood that the embodiment is not limited to the channel proportions which are shown, but it is also clear that these proportions are intended to be instructional such that at least one desirable subset of the embodiment could be construed from the diagram. Thus, an understanding of the required forces to bend a channel of these proportions will help to understand the prior art. The comparative analysis that follows uses the case of a plastic channel proportioned with lateral flanges which are 20% of its beam width.
The force required to bend a column over onto itself can be calculated by the methodology of Baumeister and Sebrosky ("Finding Vertical Column Deflections", Machine Design, Oct. 19, 1972, p159, H. K. Baumeister and R. A. Sebrosky) which uses "dimensionless column factors" to compute the loads required for large vertical deflections of vertical columns. The case of bending a 10 mil thick plastic column over onto itself is used for this analysis, and assumes the use of acetal polymer, a plastic commonly utilized for applications which require repeated flexing.
Case 1 uses the cross section of the thin strip above as made from acetal, while Case 2 uses the same material and thickness, but adds the lateral sides to form a channel with a 20% ratio of flange depth to width. This analysis shows that the rectangular section will require a bending force of 0.043 pounds (less than an ounce), while the channel shape will require a bending force of 6.38 pounds to achieve the same looped configuration. For sake of comparison, the weight of the entire translating carrier assemblage on this printer is 7 ounces. The transport motor used with a channel-shaped ink tube carrier must then be considerably larger in horsepower and physical size, and then also much more costly, than the motor required to flex the comparable flat rectangular strip. The use of a channel-shaped ink tube carrier will then be contrary to the economy and compact size goals of the desktop printer configuration, and therefore must be avoided.